analysis

Efficacy of antidepressants for adults with chronic pain

network and model

Network meta-analysis provides a means of aggregating and comparing the results of all studies examining the effect of antidepressants on substantial pain.

We now define the parameters of the model for the substantial pain outcome, using White et al.’s notation.

Let \(i = 1, \dots, n\) denote the study, and \(k = 1, \dots, K\) denote the treatment.

Let \(R_i\) be the set of treatments in study \(i\), which we call the study design.

Let \(\theta_{ik}^a\) be the parameter of interest in arm \(k\) of study \(i\) (White et al. 2019).

Let \(y_{ik}^a\) denote the observed effect for arm \(k\) of study \(i\).

arm-based likelihood

Arm-based likelihood is

contrast-based likelihood

model results

pairwise analyses

Individual analyses for each treatment, compared with placebo.

why do pairwise

pairwise meta-analysis

We want to fit a pairwise meta-analysis for each treatment, with concise summaries, as well as in-depth supplementary material. We split the analysis into two components, as the number of interventions per outcome is considerable, see below Table, in some cases, according to recommendations in Section 11.6.4 of the Handbook (Higgins et al. 2019).

Number of studies and interventions, by outcome
Interventions include placebo, as sometimes there are more than one type of placebo
Outcome Number of studies Number of interventions
substantial pain 4 5
pain 44 28
mood 61 42

pairwise model

To avoid confusion, we need to state the pairwise models being fun using the same notation as we use for the network meta-analysis models.

Here we consider a subset of data for \(j\), that is for only one outcome, \(j'\), so that we expect the same as above except we fix the index, so that \(y^a_{j = j'|ki}\).

\[ d_i \sim N(\delta_i, \sigma^2)\\ \delta_{i} \sim N(\delta, \tau^2) \] Given the data, what description of the data is most plausible, if we define the components of the data as follows? We assume a random-effects structure, which is to say, we can approximate, with a measurable degree of plausibility, the expected value of a standardised mean difference \(d_i\) of an outcome \(j'\) of interest for a study \(i\) and antidepressant treatment \(k\), as compared with placebo. We assume there is variability introduced by the study design \(\tau^2\), as well as sampling error \(\sigma^2\).

input: design matrix

A fixed effects NMA with a binomial likelihood (logit link).
Inference for Stan model: binomial_1par.
4 chains, each with iter=2000; warmup=1000; thin=1; 
post-warmup draws per chain=1000, total post-warmup draws=4000.

                   mean se_mean   sd    2.5%     25%     50%     75%
d[clomipramine]    0.46    0.00 0.26   -0.05    0.29    0.47    0.63
d[duloxetine]      0.72    0.00 0.19    0.35    0.59    0.71    0.84
d[mianserin]       0.51    0.00 0.26    0.00    0.33    0.50    0.68
d[trazodone]       0.42    0.01 0.52   -0.59    0.07    0.41    0.78
lp__            -644.48    0.05 1.99 -649.33 -645.60 -644.14 -643.03
                  97.5% n_eff Rhat
d[clomipramine]    0.96  2900    1
d[duloxetine]      1.08  2612    1
d[mianserin]       1.01  2899    1
d[trazodone]       1.46  2713    1
lp__            -641.57  1584    1

Samples were drawn using NUTS(diag_e) at Tue Apr 20 02:31:26 2021.
For each parameter, n_eff is a crude measure of effective sample size,
and Rhat is the potential scale reduction factor on split chains (at 
convergence, Rhat=1).

output

A fixed effects NMA with a binomial likelihood (logit link).
Inference for Stan model: binomial_1par.
4 chains, each with iter=2000; warmup=1000; thin=1; 
post-warmup draws per chain=1000, total post-warmup draws=4000.

                   mean se_mean   sd    2.5%     25%     50%     75%
d[clomipramine]    0.46    0.00 0.26   -0.05    0.29    0.47    0.63
d[duloxetine]      0.72    0.00 0.19    0.35    0.59    0.71    0.84
d[mianserin]       0.51    0.00 0.26    0.00    0.33    0.50    0.68
d[trazodone]       0.42    0.01 0.52   -0.59    0.07    0.41    0.78
lp__            -644.48    0.05 1.99 -649.33 -645.60 -644.14 -643.03
                  97.5% n_eff Rhat
d[clomipramine]    0.96  2900    1
d[duloxetine]      1.08  2612    1
d[mianserin]       1.01  2899    1
d[trazodone]       1.46  2713    1
lp__            -641.57  1584    1

Samples were drawn using NUTS(diag_e) at Tue Apr 20 02:31:26 2021.
For each parameter, n_eff is a crude measure of effective sample size,
and Rhat is the potential scale reduction factor on split chains (at 
convergence, Rhat=1).

structuring nma data for metafor

concise summary

long-form supplementary

Higgins, Julian PT, James Thomas, Jacqueline Chandler, Miranda Cumpston, Tianjing Li, Matthew J Page, and Vivian A Welch. 2019. Cochrane Handbook for Systematic Reviews of Interventions. John Wiley & Sons.
White, Ian R., Rebecca M. Turner, Amalia Karahalios, and Georgia Salanti. 2019. “A Comparison of Arm-Based and Contrast-Based Models for Network Meta-Analysis.” Statistics in Medicine 38 (27): 5197–5213. https://doi.org/https://doi.org/10.1002/sim.8360.

References